The more direct way to check this conclusion is by directly computing \(\mathrm {vol}(\mathbb {T})=\pi \) , using the explicit formulas for the Poincaré disk model, and then to argue that \(\mathrm {vol}(\mathbb {D})=2\pi \) , since \(\mathbb {D}\)  is the double of \(\mathbb {T}\) . Alternatively one can rely on a general version of Gauss-Bonnet formula [1]} taking into account the fact that \(\mathbb {D}\)  is homeomorphic to the sphere with three punctures and therefore it has Euler characteristic \(\chi (\mathbb {D})=-1\) . Therefore, denoting by \(\mathsf {K}_{\mathbb {D}}\)  the Gaussian curvature,
\(-\mathrm {vol}(\mathbb {D})=\int _{\mathbb {D}} \mathsf {K}_{\mathbb {D}}\,\mathsf {d}\mathrm {vol}=2\pi \chi (\mathbb {D})=-2\pi .\) 
